Monophonic Set Research Articles

Split Detour Monophonic Sets in Graph

A subset T ⊆ V is a detourmonophonic set of G if each node (vertex) x in G contained in an p-q detourmonophonic path where p, q ∈ T.. The number of points in a minimum detourmonophonic set of G is called as the detourmonophonic number of G, dm(G). A subset T ⊆ V of a connected graph G is said to be a split detourmonophonic set of G if the set T of vertices is either T = V or T is detoumonophonic set and V – T induces a subgraph in which is disconnected. The minimum split detourmonophonic set is split detourmonophonic set with minimum cardinality and it is called a split detourmonophonic number, denoted by dms(G). For certain standard graphs, defined new parameter was identified. Some of the realization results on defined new parameters were established.

On the open monophonic number of a graph

For a connected graph [Formula: see text] of order [Formula: see text], a connected open monophonic set [Formula: see text] of vertices in a connected graph [Formula: see text] is a minimal connected open monophonic set if no proper subset of [Formula: see text] is a connected open monophonic set of [Formula: see text]. The upper connected open monophonic number om[Formula: see text] of [Formula: see text] is the maximum cardinality of a minimal connected open monophonic set of [Formula: see text]. The upper connected open monophonic numbers of certain standard graphs are determined. It is proved that [Formula: see text] for the Petersen graph [Formula: see text]. Also, it is proved that for a graph [Formula: see text] of order [Formula: see text], [Formula: see text] if and only if [Formula: see text]. For positive integers [Formula: see text],[Formula: see text] and [Formula: see text] with [Formula: see text], there exists a connected graph [Formula: see text] with [Formula: see text], [Formula: see text] and [Formula: see text]. Further, it is analyzed how the addition of a pendant edge to certain standard graphs affects the upper connected open monophonic number of the respective original graphs.

The Monophonic Metric Dimension of Degree Splitting Graph

Let be a simple graph and be an ordered set and. The representation of with respect to is the -tuple . Then is called a monophonic resolving set if different vertices of have different representations with respect to . A monophonic resolving set of minimum number of elements is called a minimum monophonic set for and its cardinality is known as the monophonic metric dimension of , represented by In this article, we determined the monophonic metric dimension of degree splitting graph.

Monophonic Polynomial of the cartesian product of some graphs

Let be the family of monophonic sets of a graph with cardinality and let Then the monophonic polynomial of is defined as , where is the monophonic number of . In this paper we have determined the sufficient condition for the monophonic set of . Also, we have calculated the monophonic polynomial of the cartesian product of some specific graphs by generating function method.

The Upper Connected Outer Connected Monophonic Number of a Graph

For a connected graph of order at least two, a connected outer connected monophonic set of is called a minimal connected outer connected monophonic set if no proper subset of is a connected outer connected monophonic set of . The upper connected outer connected monophonic number of is the maximum cardinality of a minimal connected outer connected monophonic set of . We determine bounds for it and find the upper connected outer connected monophonic number of certain classes of graphs. It is shown that for any two integers with , there is a connected graph of order with and . Also, for any three integers and with , there is a connected graph with and and a minimal connected outer connected monophonic set of cardinality , where is the connected outer connected monophonic number of a graph.

Minimal connected restrained monophonic sets in graphs

For a connected graph G = (V, E) of order at least two, a connected restrained monophonic set S of G is a restrained monophonic set such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected restrained monophonic set of G is the connected restrained monophonic number of G and is denoted by mcr(G). A connected restrained monophonic set S of G is called a minimal connected restrained monophonic set if no proper subset of S is a connected restrained monophonic set of G. The upper connected restrained monophonic number of G, denoted by m+cr(G), is defined as the maximum cardinality of a minimal connected restrained monophonic set of G. We determine bounds for it and certain general properties satisfied by this parameter are studied. It is shown that, for positive integers a, b such that 4 ≤ a ≤ b, there exists a connected graph G such that mcr(G) = a and m+cr(G) = b.

On the Connected Monophonic Number of a Graph

For a connected graph G of order at least two, a connected monophonic set of G is a monophonic set S such that the subgraph induced by S is connected. The minimum cardinality of a connected monophonic set of G is the connected monophonic number of G and is denoted by . The number of extreme vertices and cut-vertices of G is its extreme-cut order . A graph G is an extreme-cut connected monophonic graph if . Some interesting results on the extreme-cut connected monophonic graphs G are studied. For positive integers r, d and with r<d, there exists an extreme-cut connected monophonic graph G with monophonic radius r, monophonic diameter d and the connected monophonic number k. Also if p, d and k are positive integers such that and , then there exists an extreme-cut connected monophonic graph G of order p with monophonic diameter d and .

Strong Total Monophonic Problems in Product Graphs, Networks, and Its Computational Complexity

Let G be a graph with vertex set as V(G) and edge set as E(G) which is simple as well as connected. The problem of strong total monophonic set is to find the set of vertices T⊆V(G), which contains no isolated vertices, and all the vertices in V(G)\T lie on a fixed unique chordless path between the pair of vertices in T. The cardinality of strong total monophonic set which is minimum is defined as strong total monophonic number, denoted by smt(G). We proved the NP‐completeness of strong total monophonic set for general graphs. The strong total monophonic number of certain graphs and networks is derived. If l, m, n are positive integers with 5 ≤ l ≤ m ≤ n and m ≤ 2l − 1, then we can construct a connected graph G with strong monophonic number l and strong total monophonic number m.

Forcing edge detour monophonic number of a graph

‎For a connected graph $G=(V,E)$ of order at least two‎, ‎an edge detour monophonic set of $G$ is a set $S$ of vertices such that every edge of $G$ lies on a detour monophonic path joining some pair of vertices in $S$‎. ‎The edge detour monophonic number of $G$ is the minimum cardinality of its edge detour monophonic sets and is denoted by $edm(G)$‎. ‎A subset $T$ of $S$ is a forcing edge detour monophonic subset for $S$ if $S$ is the unique edge detour monophonic set of size $edm(G)$ containing $T$‎. ‎A forcing edge detour monophonic subset for $S$ of minimum cardinality is a minimum forcing edge detour monophonic subset of $S$‎. ‎The forcing edge detour monophonic number $f_{edm}(S)$ in $G$ is the cardinality of a minimum forcing edge detour monophonic subset of $S$‎. ‎The forcing edge detour monophonic number of $G$ is $f_{edm}(G)=min{f_{edm}(S)}$‎, ‎where the minimum is taken over all edge detour monophonic sets $S$ of size $edm(G)$ in $G$‎. ‎We determine bounds for it and find the forcing edge detour monophonic number of certain classes of graphs‎. ‎It is shown that for every pair a‎, ‎b of positive integers with $0leq a<b$ and $bgeq 2$‎, ‎there exists a connected graph $G$ such that $f_{edm}(G)=a$ and $edm(G)=b$‎.

Minimal Total Open Monophonic Sets in Graphs

For a connected graph G of order n, a total open monophonic set S of vertices in a graph G is a minimal total open monophonic set if no proper subset of S is a total open monophonic set of G. The upper total open monophonic number of G is the maximum cardinality of a minimal total open monophonic set of G. Certain general properties regarding minimal total open monophonic sets are discussed, and also the upper total open monophonic numbers of certain standard graphs are determined. It is proved that for the Petersen graph G. Also, it is proved that for a connected graph G of order n, if and only if . For integers n and a with , , it is shown that there exists a connected graph G of order n with , and . It is also proved that for positive integers r, d and , there is a connected graph G with , and

The forcing total monophonic number of a graph

For a connected graph G = (V, E) of order at least two, a subset T of a minimum total monophonic set S of G is a forcing total monophonic subset for S if S is the unique minimum total monophonic set containing T . A forcing total monophonic subset for S of minimum cardinality is a minimum forcing total monophonic subset of S. The forcing total monophonic number ftm(S) in G is the cardinality of a minimum forcing total monophonic subset of S. The forcing total monophonic number of G is ftm(G) = min{ftm(S)}, where the minimum is taken over all minimum total monophonic sets S in G. We determine bounds for it and find the forcing total monophonic number of certain classes of graphs. It is shown that for every pair a, b of positive integers with 0 ≤ a &lt; b and b ≥ a+4, there exists a connected graph G such that ftm(G) = a and mt(G) = b.

Extreme-Support Total Monophonic Graphs

For a connected graph $$G=(V, E)$$ of order at least two, a total monophonic set of a graph $$G$$ is a monophonic set $$S$$ such that the subgraph $$G[S]$$ induced by $$S$$ has no isolated vertices. The minimum cardinality of a total monophonic set of $$G$$ is the total monophonic number of $$G$$ and is denoted by $${m}_{t}(G)$$ . The number of extreme vertices and support vertices of $$G$$ is its extreme-support order $$es(G)$$ . A graph $$G$$ is an extreme-support total monophonic graph if $${m}_{t}\left(G\right)=es(G)$$ . Some interesting results on the extreme-support total monophonic graphs $$G$$ are studied. Graphs $$G$$ with $${m}_{t}\left(G\right)=3=es(G)$$ are characterized. It is shown that for any three positive integers $$r, d$$ and $$k\ge 6$$ with $$r<d$$ , there exists an extreme-support total monophonic graph $$G$$ with monophonic radius $$r$$ , monophonic diameter $$d$$ and total monophonic number $$k$$ . Also, for any three positive integers $$d, k$$ and $$p$$ with $$2\le d\le p-5$$ and $$5\le k\le p-2$$ and $$p-d-k\ge 0$$ , there exists an extreme-support total monophonic graph $$G$$ of order $$p$$ with monophonic diameter $$d$$ and $${m}_{t}\left(G\right)=k$$ .

Connected edge rm – distance monophonic number of a graph

Let X be a graph of order at least two, which is connected. In this paper we define a new set called the connected edge rm - distance monophonic set of a graph X. Some properties of the connected edge rm -distance monophonic sets are determined. The parameter for connected edge rm - distance monophonic number is introduced. Many important results are determined and proved. Also, for every two integers a and b such that 2 ≤ a ≤ b, there exists a connected graph X with a as the edge rm -distance monophonic number and b as the connected edge rm -distance monophonic number of X is verified.

The monophonic global domination number of a graph

A set M ⊆ V is said to be a monophonic global dominating set of G if M is both a monophonic set and a global dominating set of G. The minimum cardinality of a monophonic global dominating set of G is the monophonic global domination number of G and is denoted by γm(G). A monophonic global dominating set of cardinality γm(G) is called a γm-set of G. The monophonic global domination number of certain classes of graphs are determined. It is proved that 2 ≤ γm(G) ≤ γg (G) ≤ n, where γg (G) is a geodetic global domination number of a G. It is shown that for every pair of positive integers a and b with 2 ≤ a ≤ b, there exists a connected graph G such that γm(G) = a and γg (G) = b.

The forcing monophonic and forcing geodetic numbers of a graph

&lt;p&gt;For a connected graph &lt;em&gt;G&lt;/em&gt; = (&lt;em&gt;V&lt;/em&gt;, &lt;em&gt;E&lt;/em&gt;), let a set &lt;em&gt;S&lt;/em&gt; be a &lt;em&gt;m&lt;/em&gt;-set of &lt;em&gt;G&lt;/em&gt;. A subset &lt;em&gt;T&lt;/em&gt; ⊆ &lt;em&gt;S&lt;/em&gt; is called a forcing subset for &lt;em&gt;S&lt;/em&gt; if &lt;em&gt;S&lt;/em&gt; is the unique &lt;em&gt;m&lt;/em&gt;-set containing &lt;em&gt;T&lt;/em&gt;. A forcing subset for S of minimum cardinality is a minimum forcing subset of &lt;em&gt;S&lt;/em&gt;. The forcing monophonic number of S, denoted by &lt;em&gt;fm&lt;/em&gt;(&lt;em&gt;S&lt;/em&gt;), is the cardinality of a minimum forcing subset of &lt;em&gt;S&lt;/em&gt;. The forcing monophonic number of &lt;em&gt;G&lt;/em&gt;, denoted by fm(G), is &lt;em&gt;fm&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;) = min{&lt;em&gt;fm&lt;/em&gt;(&lt;em&gt;S&lt;/em&gt;)}, where the minimum is taken over all minimum monophonic sets in G. We know that &lt;em&gt;m&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;) ≤ &lt;em&gt;g&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;), where &lt;em&gt;m&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;) and &lt;em&gt;g&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;) are monophonic number and geodetic number of a connected graph &lt;em&gt;G&lt;/em&gt; respectively. However there is no relationship between &lt;em&gt;fm&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;) and &lt;em&gt;fg&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;), where &lt;em&gt;fg&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;) is the forcing geodetic number of a connected graph &lt;em&gt;G&lt;/em&gt;. We give a series of realization results for various possibilities of these four parameters.&lt;/p&gt;

RESTRAINED DOUBLE MONOPHONIC NUMBER OF A GRAPH

For a connected graph \(G\) of order at least two, a double monophonic set \(S\) of a graph \(G\) is a restrained double monophonic set if either \(S=V\) or the subgraph induced by \(V-S\) has no isolated vertices. The minimum cardinality of a restrained double monophonic set of \(G\) is the restrained double monophonic number of \(G\) and is denoted by \(dm_{r}(G)\). The restrained double monophonic number of certain classes graphs are determined. It is shown that for any integers \(a,\, b,\, c\) with \(3 \leq a \leq b \leq c\), there is a connected graph \(G\) with \(m(G) = a\), \(m_r(G) = b\) and \(dm_{r}(G) = c\), where \(m(G)\) is the monophonic number and \(m_r(G)\) is the restrained monophonic number of a graph \(G\).

On the Strong Monophonic Number of a Graph

For a connected graph of order at least two, the strong monophonic problem is to determine a smallest set of vertices of such that, by fixing one monophonic path between each pair of the vertices of all vertices of are covered. In this paper, certain general properties satisfied by the strong monophonic sets are studied. Also, the strong monophonic number of a several families of graphs and computational complexity are determined. AMS Subject Classification: 05C12, 05C82

The Upper Complement Connected Monophonic Number of a Graph

For a connected graph a monophonic set of is said to be a complement connected monophonic set if or the subgraph is connected. The minimum cardinality of a complement connected monophonic set of is the complement connected monophonic number of and is denoted by A complement connected monophonic set in a connected graph is called a minimal complement connected monophonic set if no proper subset of is a complement connected monophonic set of . The upper complement connected monophonic number of is the maximum cardinality of a minimal complement connected monophonic set of . Some general properties under this concept are studied. The upper complement connected monophonic number of some standard graphs are determined. Some of its general properties are studied. It is shown that for any positive integers 2 ≤ a ≤b, there exists a connected graph such that ( ) = a and ( ) =b

Independent Monophonic Sets in Graphs

In this paper, we obtain two variables for the connected (p,q)-graphs G which is the minimum size of an independent monophonic set and an outer independentmonophonic set of G, termed as an independent monophonic number and an outer independent monophonic number of the connected (p,q)-graphs G, respectively.

The Total Outer Independent Monophonic Dominating Parameters in Graphs

In this paper the concept of total outer independent monophonic domination number of a graph is introduced. A monophonic set SÍV is said to be total outer independent monophonic domination set if &lt;S&gt; has no isolated vertex and &lt;V-S&gt; is an independent set.